1st Edition

Combinatorics of Spreads and Parallelisms

By Norman Johnson Copyright 2010

    Combinatorics of Spreads and Parallelisms covers all known finite and infinite parallelisms as well as the planes comprising them. It also presents a complete analysis of general spreads and partitions of vector spaces that provide groups enabling the construction of subgeometry partitions of projective spaces.

    The book describes general partitions of finite and infinite vector spaces, including Sperner spaces, focal-spreads, and their associated geometries. Since retraction groups provide quasi-subgeometry and subgeometry partitions of projective spaces, the author thoroughly discusses subgeometry partitions and their construction methods. He also features focal-spreads as partitions of vector spaces by subspaces. In addition to presenting many new examples of finite and infinite parallelisms, the book shows that doubly transitive or transitive t-parallelisms cannot exist unless the parallelism is a line parallelism.

    Along with the author’s other three books (Subplane Covered Nets, Foundations of Translation Planes, Handbook of Finite Translation Planes), this text forms a solid, comprehensive account of the complete theory of the geometries that are connected with translation planes in intricate ways. It explores how to construct interesting parallelisms and how general spreads of vector spaces are used to study and construct subgeometry partitions of projective spaces.

    Partitions of Vector Spaces
    Quasi-Subgeometry Partitions
    Finite Focal-Spreads
    Generalizing André Spreads
    The Going Up Construction for Focal-Spreads

    Subgeometry Partitions
    Subgeometry and Quasi-Subgeometry Partitions
    Subgeometries from Focal-Spreads
    Extended André Subgeometries
    Kantor’s Flag-Transitive Designs
    Maximal Additive Partial Spreads

    Subplane Covered Nets and Baer Groups
    Partial Desarguesian t-Parallelisms
    Direct Products of Affine Planes
    Jha-Johnson SL(2, q) × C-Theorem
    Baer Groups of Nets
    Ubiquity of Subgeometry Partitions

    Flocks and Related Geometries
    Spreads Covered by Pseudo-Reguli
    Flocks
    Regulus-Inducing Homology Groups
    Hyperbolic Fibrations and Partial Flocks
    j-Planes and Monomial Flocks

    Derivable Geometries
    Flocks of α-Cones
    Parallelisms of Quadric Sets
    Sharply k-Transitive Sets
    Transversals to Derivable Nets
    Partially Flag-Transitive Affine Planes
    Special Topics on Parallelisms

    Constructions of Parallelisms
    Regular Parallelisms
    Beutelspacher’s Construction of Line Parallelisms
    Johnson Partial Parallelisms

    Parallelism-Inducing Groups
    Parallelism-Inducing Groups for Pappian Spreads
    Linear and Nearfield Parallelism-Inducing Groups
    General Parallelism-Inducing Groups

    Coset Switching
    Finite E-Switching
    Parallelisms over Ordered Fields
    General Elation Switching
    Dual Parallelisms

    Transitivity
    p-Primitive Parallelisms
    Transitive t-Parallelisms
    Transitive Deficiency One
    Doubly Transitive Focal-Spreads

    Appendices
    Open Problems
    Geometry Background
    The Klein Quadric
    Major Theorems of Finite Groups
    The Diagram

    Bibliography

    Index

    Biography

    Norman L. Johnson is a professor in the Department of Mathematics at the University of Iowa.